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Full text of "Mathematical And Physical Papers - Iii"

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and their equality combined with dissymmetry, follow immediately from theory. We learn too from theory, that in order to procure ellipses it would be necessary to drill the hole in the direction of that diameter of the thermic spheroid which is conjugate to the plane of the plate.
15. Conceive a bar having a section with an arbitrary contour to be formed from an uncrystallized substance; let heat be applied in any manner at one or more places, and suppose the heat to escape again from the surface by radiation. Consider only those portions of the bar which are situated at a sufficient distance from the source or sources of heat to render insensible any irregularity arising from the mode in which the heat is communicated. If the bar be sufficiently slender, we may regard the temperature throughout a section drawn in a direction perpendicular to its length as approximately constant, without assuming thereby that the isothermal surfaces are perpendicular to the length. Let x be measured in the direction of the length, and consider the slice of the bar contained between the planes whose abscissa3 are x and x 4- dx. Let u be the temperature of the bar at the distance of the first plane, p the perimeter, and Q the area, of the section, /// the exterior conductivity, or rather the mean of the exterior conductivities in case they should vary from one gone-rating line to another; lot c, p, K, be the same as before, and put Q = ap,so that 4a is the side of a square whose area divided by its perimeter is equal to Qp~l.
The excess of the quantity of heat which enters during the time dt by the first of the piano ends of the slice over that which escapes by the second, is ultimately equal to KQtfujdtv*.dwdt. The quantity which the slice loses by radiation is ultimately equal to lipudxdt, if we take the temperature of the surrounding space, which is supposed to be constant, for the origin of temperatures. But the gain of heat by the slice is also equal, to vpQdn/dt.dMdl,. Hence we have
dn,      ,. (/"!(     li                                      .
cp ,. = A   , ,- //,.................. (17).
'  dt,         dx      a                                  '
If we suppose the heat to bo continually supplied, and the temperature to have become stationary, we get from this equation
/ h                 I h x uMv   Ka   4- Ne     /("X ............... (IS),